Introduction to Machine Learning CMU-10701 Stochastic Convergence - - PowerPoint PPT Presentation

Introduction to Machine Learning CMU-10701

Stochastic Convergence and Tail Bounds

Barnabás Póczos

Basic Estimation Theory

2

Rolling a Dice, Estimation of parameters 1,2,…,6

24 120 60 12

3

Does the MLE estimation converge to the right value? How fast does it converge?

4

Rolling a Dice Calculating the Empirical Average

Does the empirical average converge to the true mean? How fast does it converge?

5 sample traces

5

How fast do they converge?

Rolling a Dice, Calculating the Empirical Average

Key Questions

I want to know the coin parameter 2[0,1] within  = 0.1 error, with probability at least 1- = 0.95.

How many flips do I need?

6

• Do empirical averages converge?
• Does the MLE converge in the dice rolling problem?
• What do we mean on convergence?
• What is the rate of convergence?

Outline

Theory:

• Stochastic Convergences:

– Weak convergence = Convergence in distribution – Convergence in probability – Strong (almost surely) – Convergence in Lp norm

7

• Limit theorems:

– Law of large numbers – Central limit theorem

• Tail bounds:

– Markov, Chebyshev

Stochastic convergence definitions and properties

8

Convergence of vectors

9

Convergence in Distribution = Convergence Weakly = Convergence in Law

10

Notation:

Let {Z, Z1, Z2, …} be a sequence of random variables.

Definition: This is the “weakest” convergence.

Only the distribution functions converge! (NOT the values of the random variables)

1 a

11

Convergence in Distribution = Convergence Weakly = Convergence in Law

12

Continuity is important!

Proof: The limit random variable is constant 0: Example:

In this example the limit Z is discrete, not random (constant 0), although Zn is a continuous random variable. 1 1/n 1

Convergence in Distribution = Convergence Weakly = Convergence in Law

13

Properties

Scheffe's theorem:

convergence of the probability density functions ) convergence in distribution

Example:

(Central Limit Theorem)

Zn and Z can still be independent even if their distributions are the same!

Convergence in Distribution = Convergence Weakly = Convergence in Law

Convergence in Probability

14

Notation: Definition:

This indeed measures how far the values of Zn() and Z() are from each other.

Almost Surely Convergence

15

Notation: Definition:

Convergence in p-th mean, Lp norm

Definition:

16

Notation: Properties:

Counter Examples

17

Further Readings on Stochastic convergence

• http://en.wikipedia.org/wiki/Convergence_of_random_variables
• Patrick Billingsley: Probability and Measure
• Patrick Billingsley: Convergence of Probability Measures

18

Finite sample tail bounds

Useful tools!

19

Gauss Markov inequality

Decompose the expectation

If X is a nonnegative random variable and a > 0, then

Proof: Corollary: Chebyshev's inequality

20

Chebyshev inequality

If X is any nonnegative random variable and a > 0, then

Proof: Here Var(X) is the variance of X, defined as:

21

Generalizations of Chebyshev's inequality

Chebyshev:

Asymmetric two-sided case (X is asymmetric distribution) Symmetric two-sided case (X is symmetric distribution) This is equivalent to this: There are lots of other generalizations, for example multivariate X.

22

Higher moments?

Chebyshev: Markov: Higher moments:

where n ≥ 1

Other functions instead of polynomials? Exp function: Proof:

(Markov ineq.)

23

Law of Large Numbers

24

Do empirical averages converge?

25

Answer: Yes, they do. (Law of large numbers)

Chebyshev’s inequality is good enough to study the question:

Do the empirical averages converge to the true mean?

Law of Large Numbers

26

Strong Law of Large Numbers: Weak Law of Large Numbers:

Weak Law of Large Numbers

Proof I:

Assume finite variance. (Not very important) Therefore, As n approaches infinity, this expression approaches 1.

27

Fourier Transform and Characteristic Function

28

Fourier Transform

29

Fourier transform Inverse Fourier transform

Other conventions: Where to put 2?

Not preferred: not unitary transf. Doesn’t preserve inner product unitary transf. unitary transf.

Fourier Transform

30

Fourier transform Inverse Fourier transform

Inverse is really inverse:

Properties:

and lots of other important ones…

Fourier transformation will be used to define the characteristic function, and represent the distributions in an alternative way.

Characteristic function

31

The Characteristic function provides an alternative way for describing a random variable

How can we describe a random variable?

• cumulative distribution function (cdf)
• probability density function (pdf)

Definition: The Fourier transform of the density

Characteristic function

32

Properties

For example, Cauchy doesn’t have mean but still has characteristic function. Continuous on the entire space, even if X is not continuous. Bounded, even if X is not bounded Levi’s: continuity theorem Characteristic function of constant a:

Weak Law of Large Numbers

Proof II:

33

Taylor's theorem for complex functions The Characteristic function

Properties of characteristic functions : Levi’s continuity theorem ) Limit is a constant distribution with mean  mean

34

“Convergence rate” for LLN

Gauss-Markov: Doesn’t give rate Chebyshev:

Can we get smaller, logarithmic error in δ???

with probability 1-

• http://en.wikipedia.org/wiki/Levy_continuity_theorem
• http://en.wikipedia.org/wiki/Law_of_large_numbers
• http://en.wikipedia.org/wiki/Characteristic_function_(probability_theory)
• http://en.wikipedia.org/wiki/Fourier_transform

35

Further Readings on LLN, Characteristic Functions, etc

More tail bounds

More useful tools!

36

Hoeffding’s inequality (1963)

It only contains the range of the variables, but not the variances.

37

“Convergence rate” for LLN from Hoeffding

Hoeffding

38

Proof of Hoeffding’s Inequality

39

A few minutes of calculations.

Bernstein’s inequality (1946)

It contains the variances, too, and can give tighter bounds than Hoeffding.

40

Benett’s inequality (1962)

Benett’s inequality ) Bernstein’s inequality.

41

Proof:

McDiarmid’s Bounded Difference Inequality

It follows that

42

Further Readings on Tail bounds

43

http://en.wikipedia.org/wiki/Hoeffding's_inequality http://en.wikipedia.org/wiki/Doob_martingale (McDiarmid) http://en.wikipedia.org/wiki/Bennett%27s_inequality http://en.wikipedia.org/wiki/Markov%27s_inequality http://en.wikipedia.org/wiki/Chebyshev%27s_inequality http://en.wikipedia.org/wiki/Bernstein_inequalities_(probability_theory)

Limit Distribution?

44

Central Limit Theorem

45

Lindeberg-Lévi CLT: Lyapunov CLT:

+ some other conditions Generalizations: multi dim, time processes

Central Limit Theorem in Practice

unscaled scaled

46

47

Proof of CLT

From Taylor series around 0: Properties of characteristic functions : Levi’s continuity theorem + uniqueness ) CLT

characteristic function

• f Gauss distribution

How fast do we converge to Gauss distribution?

Berry-Esseen Theorem

48

Independently discovered by A. C. Berry (in 1941) and C.-G. Esseen (1942)

CLT:

It doesn’t tell us anything about the convergence rate.

Did we answer the questions we asked?

• Do empirical averages converge?
• What do we mean on convergence?
• What is the rate of convergence?
• What is the limit distrib. of “standardized” averages?

 How good are the ML algorithms on unknown test sets?  How many training samples do we need to achieve small error?  What is the smallest possible error we can achieve?

Next time we will continue with these questions:

49

• http://en.wikipedia.org/wiki/Central_limit_theorem
• http://en.wikipedia.org/wiki/Law_of_the_iterated_logarithm

Further Readings on CLT

50

Tail bounds in practice

51

• Two possible webpage layouts
• Which layout is better?

Experiment

• Some users see A
• The others see design B

How many trials do we need to decide which page attracts more clicks?

A/B testing

52

A/B testing

53

Assume that in group A p(click|A) = 0.10 click and p(noclick|A) = 0.90 Assume that in group B p(click|B) = 0.11 click and p(noclick|B) = 0.89 Assume also that we know these probabilities in group A, but we don’t know yet them in group B.

We want to estimate p(click|B) with less than 0.01 error Let us simplify this question a bit:

• In group B the click probability is  = 0.11 (we don’t know this yet)
• Want failure probability of =5%

Chebyshev Inequality

54

Chebyshev:

• If we have no prior knowledge, we can only bound the variance by σ2 =

0.25 (Uniform distribution hast the largest variance 0.25)

• If we know that the click probability is < 0.15, then we can bound 2 at

0.15 * 0.85 = 0.1275. This requires at least 25,500 users.

Hoeffding’s bound

• Random variable has bounded range [0, 1] (click or no click),

hence c=1

• Solve Hoeffding’s inequality for n:

55

This is better than Chebyshev.

• Hoeffding

What we have learned today

56

Theory:

• Stochastic Convergences:

– Weak convergence = Convergence in distribution – Convergence in probability – Strong (almost surely) – Convergence in Lp norm

• Limit theorems:

– Law of large numbers – Central limit theorem

• Tail bounds:

– Markov, Chebyshev – Hoeffding, Benett, McDiarmid

Thanks for your attention 

57